Your MOM's a determinant

To my sister and her classmates, so that their math homework can be just a little less depressing.

Here are a few perfectly reasonable questions from high school sophomores about finding the determinant of a matrix:

• What are they good for?
• Who cares?
• This is stupid and I hate you.

Let's address these questions. The first stop in the 21st century is, of course, Wikipedia:

The determinant provides important information about a matrix of coefficients of a system of linear equations, or about a matrix that corresponds to a linear transformation of a vector space. In the first case the system has a unique solution exactly when the determinant is nonzero; when the determinant is zero there are either no solutions or many solutions. In the second case the transformation has an inverse operation exactly when the determinant is nonzero. A geometric interpretation can be given to the value of the determinant of a square matrix with real entries: the absolute value of the determinant gives the scale factor by which area or volume (or a higher-dimensional analogue) is multiplied under the associated linear transformation, while its sign indicates whether the transformation preserves orientation. Thus a 2 × 2 matrix with determinant −2, when applied to a region of the plane with finite area, will transform that region into one with twice the area, while reversing its orientation.
Determinants occur throughout mathematics. The use of determinants in calculus includes the Jacobian determinant in the substitution rule for integrals of functions of several variables. They are used to define the characteristic polynomial of a matrix that is an essential tool in eigenvalue problems in linear algebra. In some cases they are used just as a compact notation for expressions that would otherwise be unwieldy to write down.

Ugh, I think I see the problem. These are, actually, really good reasons to be familiar with determinants, but most of these concepts are held back until a college-level Linear Algebra course. On that day when these sophomores take Linear Algebra, they will realize the brilliance of determinants--but they're depressed now. Let's try to do better than that.

A matrix's determinant tells you

• whether a matrix is invertible, which tells you
• whether you can undo a matrix multiplication operation
• whether you can solve a system of equations based on that matrix
• what happens to a vector when it's multiplied by the matrix:
• How much does it stretch?
• Does it flip inside out?
• There could also be some rotation of the vector, but if we only care about its size and inside-outness, we can save the trouble of matrix multiplication if we know the determinant.

This, of course, presupposes that you care about matrix multiplication, which is itself a worthy topic for a similar post.

In addition to what a determinant tells you about a matrix, there are a few mathematical formulas that can be succinctly represented as computing the determinant of a matrix. This could save space on a cheat sheet for a future physics test. This is a nice collection of determinants doing cool things.

Look, this determinant is doing volume!

The procedure for computing the determinant of a 3x3 or larger matrix illustrates a key idea in mathematics: recursion. You find the determinant of a large matrix by finding the determinants of smaller submatrices and combining the results. This pattern of a procedure for solving a problem including running the procedure against a smaller version of the current problem is called recursion and pops up everywhere once you start looking for it.

The most important thing you get out of studying determinants in high school...

Drumroll, please.

It's tedious and awful.

You're insane.

Let me explain. Why is it important for high school students to be tortured with tedious, awful plug-n-chug work? Because it's a strong motivator for the self-study of computer programming.

That came out of nowhere.

Let me explain.

If you're in high school today, you've never known a planet without personal computers. You probably spend more time on tablets than you do watching TV. More and more data about what you do every day are winding up in computer systems. More and more jobs require the ability to analyze and manipulate data. Literacy was to the 20th century as digital literacy is to the 21st. Until computer programming is part of a normal curriculum, you're on your own to learn this stuff.

Computing the determinant of a large matrix is the sort of "hard" that mankind invented computers to solve: lots of simple steps strung together. You won't be hired to find the determinant of a matrix because we have software that does that. There are lots of other computational activities taken over by computers as well--pretty much any well-defined useful task that can be broken down into simple steps. This is a good thing because it lets people focus their time and energy on the next problem (there's always a next problem). The lessons you learn writing software to help you cheat on your math homework will be applied again and again throughout your career.

Knowing all this won't make finding determinants less tedious, but hopefully it won't seem like entirely random torture. Good luck!